4,316 research outputs found
Primes in short arithmetic progressions
We give a large sieve type inequality for functions supported on primes. As
application we prove a conjecture by Elliott, and give bounds for short
character sums over primes. The proves uses a combination of the large sieve
and the Selberg sieve
Primes in short arithmetic progressions
Let and be three parameters. We show that, for most moduli
and for most positive real numbers , every reduced arithmetic
progression has approximately the expected number of primes from
the interval , provided that and satisfies
appropriate bounds in terms of and . Moreover, we prove that, for most
moduli and for most positive real numbers , there is at least
one prime lying in every reduced arithmetic progression , provided that .Comment: 21 pages. Final version, published in IJNT. Some minor change
Large gaps between consecutive prime numbers
Let denote the size of the largest gap between consecutive primes
below . Answering a question of Erdos, we show that where
is a function tending to infinity with . Our proof combines existing
arguments with a random construction covering a set of primes by arithmetic
progressions. As such, we rely on recent work on the existence and distribution
of long arithmetic progressions consisting entirely of primes.Comment: v2. very minor corrections. To appear in Ann. Mat
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